GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 04 Apr 2020, 14:26

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

# For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

Author Message
TAGS:

### Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 62499
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

23 Aug 2015, 21:49
9
59
00:00

Difficulty:

65% (hard)

Question Stats:

59% (02:04) correct 41% (01:49) wrong based on 631 sessions

### HideShow timer Statistics

For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 10240
Location: Pune, India
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

08 Jan 2016, 22:07
10
1
20
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

This is a property of exponents:

$$2^1 + 2^1 = 2^2$$
$$2^2 + 2^2 = 2^3$$ (because $$2^2 * (1 + 1) = 2^2 * 2$$)
Similarly, $$2^3 + 2^3 = 2^4$$
$$2^4 + 2^4 = 2^5$$
etc

So
$$2^{29} + 2^{29} = 2^{30}$$

$$x + y = 29 + 29 = 58$$

Similarly,

$$3^1 + 3^1 + 3^1 = 3^2$$
$$3^2 + 3^2 + 3^2 = 3^3$$
$$3^3 + 3^3 + 3^3 = 3^4$$
and so on...
_________________
Karishma
Veritas Prep GMAT Instructor

Manager
Joined: 15 May 2014
Posts: 61
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

24 Aug 2015, 20:38
10
11
$$2^x+2^y\,=\,2^{30}$$
$$x+y\,=\,?$$

$$2^x\,(1+2^{y-x})\,=\,2^{30}$$
$$1+2^{y-x}\,=\,2^{30-x}$$
$$1\,=\,2^{30-x}-2^{y-x}$$

Difference between any two powers of 2 yield 1 if one of the powers is one and the other is zero
i.e. $$2^1\,-\,2^0\,=\,1$$

--> $$30-x\,=\,1\,\,\,$$ and $$\,\,\,y-x\,=\,0$$
$$x\,=\,29\,\,\,$$ and $$\,\,\,x\,=\,y$$
$$x+y\,=\,58$$

##### General Discussion
Intern
Joined: 28 Jan 2013
Posts: 31
Location: United States
GPA: 3.1
WE: Information Technology (Consulting)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

23 Aug 2015, 22:09
4
7
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

1/2 * (2^30 + 2^30) = 2^30

2^29 + 2^29 = 2^30

The above equation is similar to 2^x + 2^y =2^30.
so x =29 and y=29

Ans:D
Current Student
Joined: 21 Jan 2015
Posts: 453
Location: India
Concentration: Strategy, Marketing
GMAT 1: 620 Q48 V28
GMAT 2: 690 Q49 V35
WE: Sales (Consumer Products)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

23 Aug 2015, 22:22
6
5
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Ans: D

Solution: 2^x + 2^y =2^30
there must be some value of as we know 2^30 only has 2 as its factor so if i take anything common from LHS of the equation the remaining part must be in form of 2 only.
I can take common if and only if x=y; otherwise the remaining part inside the bracket will become odd
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y
so now=>
the equation becomes 2^x + 2^x =2^30 because x=y
2^x (1+1)= 2^30
2^x * 2 = 2^30
x+1 = 30
x= 29
and y = 29
x+y = 58
_________________
--------------------------------------------------------------------
The Mind is Everything, What we Think we Become.
Manager
Joined: 14 Mar 2014
Posts: 140
GMAT 1: 710 Q50 V34
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

23 Aug 2015, 23:48
6
4
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

IMO: D

Both sides consists of only powers of 2.

Thus in order to that happen x = y must be the case

if x=y then

$$2^x +2^y =2 ^30$$ ==> $$2^x +2^x =2 ^30$$
$$2(2^x) =2 ^30$$
x+1 =30
x=29
Thus x+y = 29+29 =58
Manager
Joined: 15 May 2014
Posts: 61
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

25 Aug 2015, 00:43
5
2
2^1 = 2^0 + 2^0
2^2 = 2^1 + 2^1
2^3 = 2^2 + 2^2
.
.
.
2^30 = 2^29 + 2^29

x + y = 58

Option D
Director
Joined: 21 May 2013
Posts: 622
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

25 Aug 2015, 11:15
1
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

2^x+2^y=2^30
Now, 2^30=2^29+2^29=2*2^29
or x+y=29+29=58
Intern
Joined: 02 Jun 2015
Posts: 32
Location: United States
Concentration: Operations, Technology
GMAT Date: 08-22-2015
GPA: 3.92
WE: Science (Other)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

25 Aug 2015, 13:11
dkumar2012 wrote:
for example lets say x= 14 and y = 16 then 2^14+2^16 = 2^14 (1+2^2)= 2^14 * 5 .. and this will happen for all the values of x and y where x is not equal to y

This is a great solution. A little more detail for those wanting to see why all cases will not work (not just the one above)

2^x + 2^y = 2^x * (1+2^(y-x)) = 2^30

This means that 1+2^(y-x) has to be a multiple of 2 and this can only be achieved if 2^(y-x) = 1 = 2^0.
Math Expert
Joined: 02 Sep 2009
Posts: 62499
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

30 Aug 2015, 07:41
3
6
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION:

This problem is a good candidate for testing small numbers to find patterns. While your instincts may tell you to simply set x + y equal to 30, small numbers will show that you cannot simply set them equal. If you try $$2^x+2^y=2^6$$, for example you can't find values for x + y = 6 that will set the sum equal to 64. The powers of 2 are:

2

4

8

16

32

64

The only pairing that will work is 32 + 32 = 64, meaning that you need $$2^5+2^5=2^6$$, which should make sense: 2 times 2^5 will equal 2^6. So this example should teach you that you need to add two $$2^{29}$$s together to get to $$2^{30}$$. So x and y are each 29, making the sum x + y equal to 58, answer choice D.
_________________
Intern
Joined: 03 Dec 2014
Posts: 14
GMAT 1: 620 Q43 V32
GPA: 2.9
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

19 Nov 2015, 16:58
Bunuel wrote:
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

2^x+2^Y = 2^30

2^x-30 + 2^y-30 = 1 (Divide LHS and RHS by 2^30)

1/2+1/2 = 1 Therefore 2^x-30=1/2 i.e. x-30 = -1 hence X= 29
similarly y-30 = -1 and y = 29

X+Y = 29+29 = 58

Ans D
CEO
Joined: 20 Mar 2014
Posts: 2542
Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

08 Jan 2016, 09:08
1
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

You are given 2^x+2^y=2^30 ---> in order to understand the question, try to see what values can y take if you start with fixed values for x ---> remember that x and y MUST be integers.

Thus, lets have x =1 ---> $$2^y=2^{30}-2$$ ---> y can not be an integer. Try a bigger value for x.

x = 15 ---> $$2^y=2^{30}-2^{15}$$ ---> $$2^y=2^{15}(2^{15}-1)$$ , keep trying and you will see that there will always be a 1 inside the bracket except when you write the given expression as

$$2^x+2^y=2^{30}$$ --->$$2^x+2^y=2*2^{29}$$ ---> $$2^x+2^y=2^{29}+2^{29}$$ ---> compare both sides of the equation to get x=y=29 and x+y = 58.

D is thus the correct answer.

Hope this helps.
Math Expert
Joined: 02 Aug 2009
Posts: 8308
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

11 Jan 2016, 08:12
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

Hi,
here we are adding two different numbers which are power to base 2 resulting into another number which has a base of 2..
the logic is .. if we are adding 2^x and 2^y to get 2^z, only way is to get 2 out of the addition and that is possible only when x and y are same..
so $$2^x+2^y=2^{30}$$..where x =y
can be written as $$2^x+2^x=2^{30}$$
$$2*2^x=2^{30}$$..
$$x+1=30 ..x=29$$..
so $$x+y=29+29=58$$
_________________
Current Student
Status: Gaja!
Joined: 26 Aug 2014
Posts: 49
Location: United States (CA)
GMAT 1: 700 Q49 V36
GMAT 2: 680 Q49 V34
GMAT 3: 740 Q50 V40
GPA: 3.6
WE: Consulting (Consulting)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

16 Apr 2016, 18:28
1
Here's a better way to solve the problem (suggested by Anthony Ritz - big thank you! ).

Step 1: factor out the given equation and see if there is a pattern
$$2^x(1 + 2^{y-x}) = 2^{30}$$
Hm, interesting. We know that $$2^x$$ is always even, so $$(1 + 2^{y-x})$$ should also be even since $$2^{30}$$ is even.

For $$(1 + 2^{y-x})$$ to be an even integer, $$2^{y-x}$$ needs to be odd. What situation will $$2^{y-x}$$ be an odd number? When $$2^{y-x} = 1$$!
Hence,$$y-x = 0$$ and consequently $$x=y$$.

Step 2: plug in our findings to the equation
$$2^x + 2^y = 2^x + 2^x = 2 * 2^x = 2^{1+x} = 2^{30}$$
The last two parts of our equation tells us that $$1+x = 30$$.
In conclusion, $$x = y = 29$$.

Answer: $$x + y = 58$$.
Manager
Joined: 10 Apr 2016
Posts: 54
Concentration: Strategy, Entrepreneurship
GMAT 1: 520 Q29 V30
GPA: 3.01
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

17 Apr 2016, 02:42
1
It's actually really easy to spot a pattern here.

2^2 + 2^2 = 8
2^3 = 8
2^4 + 2^4= 16 + 16 = 32
2^5 = 32

So 2^x +2^y = 2^n
to get x+y you just do 2(n-1).
_________________
Took the Gmat and got a 520 after studying for 3 weeks with a fulltime job. Now taking it again, but with 6 weeks of prep time and a part time job. Studying every day is key, try to do at least 5 exercises a day.
Current Student
Status: DONE!
Joined: 05 Sep 2016
Posts: 340
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

05 Nov 2016, 17:35
I used a simple tester example to see what would happen if I wanted to find a smaller exponential value of 2.

I used the following:

2^4 = 2^x + 2^y --> 8+8 (i.e. 2^3 + 2^3) satisfies this...

sum of x and y has to be one less than exponent we are looking for. Thus 29+29 = 58 and we have our answer.
Manager
Joined: 19 Oct 2016
Posts: 63
Location: India
Schools: IIMA (I)
GMAT 1: 580 Q46 V24
GMAT 2: 540 Q39 V25
GMAT 3: 660 Q48 V34
GPA: 3.15
WE: Psychology and Counseling (Health Care)
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

05 Nov 2016, 22:34
Bunuel wrote:
For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

Love this as a 700 level question. Easy if you keep your head on straight. Plug in 2^2 + 2^2= 8 = 2^3.... Plug in 2^3 +2^3= 16 = 2^4....As we can see the pattern shows that 2^29 + 2 ^29 will equal 2^30. Then we can add 29 + 29 and get 58.

Under 30 seconds oooooooooooooooooooooh yea
Intern
Joined: 11 Dec 2016
Posts: 47
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

11 May 2018, 04:32
1
VeritasPrepKarishma wrote:
For integers x and y, 2^x+2^y=2^30. What is the value of x+y?

A. 30
B. 32
C. 46
D. 58
E. 64

This is a property of exponents:

$$2^1 + 2^1 = 2^2$$
$$2^2 + 2^2 = 2^3$$ (because $$2^2 * (1 + 1) = 2^2 * 2$$)
Similarly, $$2^3 + 2^3 = 2^4$$
$$2^4 + 2^4 = 2^5$$
etc

So
$$2^{29} + 2^{29} = 2^{30}$$

$$x + y = 29 + 29 = 58$$

Similarly,

$$3^1 + 3^1 + 3^1 = 3^2$$
$$3^2 + 3^2 + 3^2 = 3^3$$
$$3^3 + 3^3 + 3^3 = 3^4$$
and so on...

Thanks for making things much easier!! Love your blogs as well. Keep up the great work.
Senior Manager
Joined: 29 Dec 2017
Posts: 365
Location: United States
Concentration: Marketing, Technology
GMAT 1: 630 Q44 V33
GMAT 2: 690 Q47 V37
GMAT 3: 710 Q50 V37
GPA: 3.25
WE: Marketing (Telecommunications)
For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

11 May 2018, 12:42
Alternative approach if you don's see the pattern:

$$2^x + 2^y=2^{30}$$, where $$2^x + 2^y$$=$$(2^x +2^y)^2-2*2^{x+y}=2^{30}$$, where $$(2^x + 2^y)^2=2^{60}$$

$$2^{30}*(2^{30}-1)=2^{x+y+1}$$ ,

we can omit -1 since it almost doesn't change the result, so $$2^{60}= 2^{x+y+1}$$, x+y+1=60. x+y = 59 - yes I know that this is not 58, but this is the closest result. Answer (D)
Director
Joined: 14 Dec 2017
Posts: 505
Location: India
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?  [#permalink]

### Show Tags

08 Jun 2018, 01:06
Bunuel wrote:
For integers x and y, $$2^x + 2^y=2^{30}$$. What is the value of $$x + y$$?

A. 30
B. 32
C. 46
D. 58
E. 64

Kudos for a correct solution.

$$2^x + 2^y=2^{30}$$
which can be written as, $$2^{29}*({2^{x-29}} + {2^{y-29}})=2^{30}$$

Hence, $${2^{x-29}} + {2^{y-29}} = 2$$

Since x & y are integers, we got $$x-29 = 0$$ & $$y-29 = 0$$

$$x = 29$$, $$y =29$$, $$x+y = 58$$

Thanks,
GyM
_________________
Re: For integers x and y, 2^x + 2^y =2^30. What is the value of x + y?   [#permalink] 08 Jun 2018, 01:06

Go to page    1   2    Next  [ 25 posts ]

Display posts from previous: Sort by